Dopesheet - What is the difference between Select methods Border Axis Range and Border select?
Old question, but the answer might be useful to someone:


Border Select: Selects the elements inside of a given rectangle.
Border Axis Range: Selects the elements between the left and right sides of the rectangle, regardless of their vertical position.


Example:


In other wordes, Border Axis Range is a range selection tool, while Border Select is a box selection tool. 

Border Axis Range is useful for armatures with a large number of bones: If you cannot fit all the rows in one window, you can still select over a range without having to scroll up or down.